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Portfolio Construction and Analytics by Dessislava A. Pachamanova, Frank J. Fabozzi

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Chapter 7Optimization under Uncertainty

The optimization framework presented in Chapter 6 has numerous applications in finance and other fields. However, our discussion omitted an important aspect of realistic optimization modeling. We assumed that the input data, such as the coefficients in front of the decision variables in the objective function and the constraints, are certain. In practice, however, optimization often needs to be performed under conditions in which the input data are random, or represent statistical estimates and subjective guesses. Models in which all input data are fixed, or nonrandom, are referred to as deterministic. Models that contain parameters that vary are referred to as nondeterministic, probabilistic, or stochastic.

Concepts from probability theory, statistics, and simulation (Chapters 2, 3, and 5) can be used to extend the basic framework of deterministic optimization to deal with uncertainty. Randomness, however, adds a high level of complexity to optimization formulations, and the output of the resulting models needs to be interpreted carefully.

There are three general approaches for incorporating uncertainty into optimization problems: dynamic programming, stochastic programming, and robust optimization. Dynamic programming methods date back to Bellman (1957) and are specifically designed to deal with stochastic uncertain systems over multiple stages. The optimization problem is solved recursively, going backwards from the last state, and ...

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